SPECIAL SESSION 22 95

Special Session 22: Topological and Variational Methods for BoundaryValue Problems

John R. Graef, University of Tennessee at Chattanooga, USALingju Kong, University of Tennessee at Chattanooga, USA

Bo Yang, Kennesaw State University, USA

Topological methods have proved to be an important technique in the study of boundary value problemsand related topics for ordinary and partial di↵erential equations. Recently that has been a rapidly growinginterest in applying variational methods and critical point theory to such problems. This session is devotedto the use of these methods in the study of boundary value problems including singular problems and thosewith multipoint conditions.

Existence of multiple positive solutions for p-Laplacian multipoint boundary value problemon time scales

Abdulkadir DoganAbdullah Gul University, Turkeyabdulkadir94@hotmail.com

In this paper, we consider p-Laplacian multipointboundary value problem on time scales. By usingfixed point theorems, we prove the existence of atleast three positive solutions to the boundary valueproblem. The interesting point is that the nonlinearterm f depends on the first order derivative explicity.As an application, an example is given to illustratethe result.

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A note on a third-order multi-point boundaryvalue problem at resonance

Zengji DuXuzhou Normal University, Peoples Rep of Chinaduzengji@163.comXiaojie Lin, Fanchao Meng

Based on the coincidence degree theory of Mawhin,we prove some existence results for a third-ordermulti-point boundary value problem at resonance.In this talk, the dimension of the linear space KerL is equal to 2. Since all the existence results forthird-order di↵erential equations obtained in previ-ous papers are for the case dim Ker L = 1, our workis new.

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A class of decomposable nonlinear operatorsand its applications in BVP

Wenying FengTrent University, Canadawfeng@trentu.caJianhong Wu

We will introduce a class of nonlinear operators thatcan be decomposed into a linear operator and anonlinear map. Some properties for the class areproved. Applications to existence of solutions for

some boundary value problems are given.

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Generalized upper and lower solutions onfourth order Lidstone problems

Joao FialhoCIMA - UE, Portugaljfzero@gmail.comFeliz Minhos

In this work it is considered the nonlinear fully equa-tion

u(iv) (x) + f�x, u (x) , u0 (x) , u00 (x) , u000 (x)

�= sp (x)

(1)for x 2 [0, 1] , where f : [0, 1] ⇥ R4 ! R andp : [0, 1] ! R+ are continuous functions and s areal parameter, coupled with the Lidstone boundaryconditions,

u(0) = u(1) = u00(0) = u00(1) = 0, (2)

These types of problems are known as Ambrosetti-Prodi problems, and they provide the discussion ofexistence, nonexistence and multiplicity results onthe parameter s. More precisely, su�cient conditions,for the existence of s0 and s1, are obtained, such that:

• if s < s0, the problem has no solution.

• if s = s0, the problem has a solution.

• if s 2 ]s0, s1] , the problem has at least twosolutions.

In this work it is discussed how conditions in thelower and upper definitions influence the main resultsand vice-versa. This ”power shift” between the Defi-nition and Theorem makes it possible to extend someresults to a functional version of (1)-(2). In additionwe replace the usual bilateral Nagumo condition by aone-sided condition, allowing the nonlinearity to beunbounded.

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96 9th AIMS CONFERENCE – ABSTRACTS

On a notion of category depending on a func-tional and an application to Hamiltonian sys-tems

Marlene FrigonUniversity of Montreal, Canadafrigon@dms.umontreal.ca

A notion of category depending on a functional isintroduced. This notion permits to obtain a betterlower bound on the number of critical points of afunctional than the classical Lusternik-Schnirelmancategory. A relation between this notion and thelinking of type splitting spheres is discussed. An ap-plication to Hamiltonian systems is also presented.

References: [1] N. Beauchemin and M. Frigon, On anotion of category depending on a functional. PartII: An application to Hamiltonian systems, NonlinearAnal. 72 (2010) 3376–3387. [2] N. Beauchemin andM. Frigon, On a notion of category depending on afunctional. Part I: Theory and Application to criticalpoint theory, Nonlinear Anal. 72 (2010) 3356–3375.[3] C.C. Conley and E. Zehnder, The Birkho↵-Lewisfixed point theorem and a conjecture of V.I. Arnol’d,Invent. Math. 73 (1983), 33–49.[4] G. Fournier, D.Lupo, M. Ramos and M. Willem, Limit relative cate-gory and critical point theory, in Dynamics reported.Expositions in Dynamical Systems, vol. 3, 1–24,Springer, Berlin, 1994.[5] M. Frigon, On a new no-tion of linking and application to elliptic problemsat resonance, J. Di↵erential Equations, 153 (1999),96–120.[6] A. Szulkin, A relative category and appli-cations to critical point theory for strongly indefinitefunctionals, Nonlinear Anal. 15 (1990), 725-739.

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Existence of nontrivial solutions to systems ofmulti-point boundary value problems

John GraefUniversity of Tennessee at Chattanooga, USAjohn-graef@utc.eduLingju Kong, Shapour Heidarkhani

The authors consider the system of n multi-pointboundary value problems

(�(�pi(u

0i))

0 = �Fui(x, u1, ..., un), x 2 (0, 1),ui(0) =

Pmj=1 ajui(xj), ui(1) =

Pmj=1 bjui(xj),

for i = 1, . . . , n. The existence of at least one non-trivial solution is proved using variational methodsand critical point theory.

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Existence and multiplicity for positive solu-tions of a system of higher-order multi-pointboundary value problems

Johnny HendersonBaylor University, USAJohnny Henderson@baylor.eduRodica Luca

We investigate the existence and multiplicity of pos-itive solutions of multi-point boundary value prob-lems for systems of nonlinear higher-order ordinarydi↵erential equations.

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Solvability of second order three-point bound-ary value problem at resonance

James S. W. WongThe University of Hong Kong, Hong Kongjsww@chinneyhonkwok.comC. H. Ou

We are interested in existence theorems for secondorder three-point boundary value problems at res-onance. The usual method of proof is based uponcoincidence degree theory which di↵ers from thefixed point theorem approach for problems subjectto non-resonant boundary conditions. We introducean alternate method by reducing the original prob-lem to a two-point problem with non-homogeneousboundary condition containing a parameter µ. Wethen apply fixed point theorem and shooting methodto determine the proper µ which gives rise to a solu-tion of the original three-point problem.

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On a discrete fourth order periodic boundaryvalue problem

Lingju KongUniversity of Tennessee at Chattanooga, USALingju-Kong@utc.eduJohn R. Graef, Min Wang

By using the variational method and critical pointtheory, we obtain criteria for the existence of mul-tiple solutions of the discrete fourth order periodicboundary value problem

�4u(t� 2)� ↵�2u(t� 1) + �u(t) = f(t, u(t)), t 2 [1, T ]Z,�iu(�1) = �iu(T � 1), i = 0, 1, 2, 3,

where T � 2 is an integer, [1, T ]Z = {1, 2, . . . , T},↵,� � 0 are parameters, and f : [1, T ]Z ⇥ R ! Ris a continuous function. Examples are included toillustrate the results.

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SPECIAL SESSION 22 97

A fourth-order functional problem at reso-nance

Nickolai KosmatovUniversity Of Arkansas at little Rock, USAnxkosmatov@ualr.edu

We study the di↵erential equation

u(4)(t)� !4u(t) = f(t, u(t), u0(t), u00(t), u000(t)),

satisfying linear functional conditions

Biu = 0, i = 1, ..., 4.

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Boundary data smoothness for solutions ofnth order nonlocal boundary value problems

Je↵rey LyonsTexas A&M University - Corpus Christi, USAje↵.lyons@tamucc.edu

In this talk, we investigate boundary data smooth-ness for solutions of the nonlocal boundary valueproblem, y(n) = f(x, y, y0, . . . , y(n�1)), y(i)(xj) = yij

and y(i)(xk) �mX

p=1

ripy(⌘ip) = yik. Essentially, we

show under certain conditions that partial deriva-tives of the solution to the problem above exist withrespect to boundary conditions and solve the asso-ciated variational equation. Lastly, there will be acorollary and nontrivial example.

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Two-point boundary value problems with im-pulses.

Daniel MaroncelliNorth Carolina State University, USAdmmaronc@ncsu.eduJesus Rodrıguez

The authors look at nonlinear boundary value prob-lems of the form

x0(t) = A(t)x(t) + f(t, x(t)), t 2 [0, 1] \ {t1, · · · , tk}

x(t+i )� x(ti) = Ji(x(ti)), i = 1, ..., k

subject toBx(0) +Dx(1) = 0.

We focus on the resonant case, that is, the case inwhich the solution space of the linear homogeneousproblem is nontrivial. In particular, we use degreetheory to prove the existence of solutions when thesolution space has dimension greater than 1.

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Existence and multiplicity of solutions infourth order BVPs with unbounded nonlin-earities

Feliz MinhosUniversity of Evora, Portugalfminhos@uevora.ptJ. Fialho

This work studies the Ambrosetti-Prodi fourth ordernonlinear fully equation

u(4) (x) + f�x, u (x) , u0 (x) , u00 (x) , u000 (x)

�= s p(x)

for x 2 [a, b] , f : [a, b] ⇥ R4 ! R, p : [a, b] ! R+

continuous functions and s 2 R, with the boundaryconditions

u(a) = A, u0(b) = B, u000(a) = C, u000(b) = D.

In this work it will be presented an Ambrosetti-Proditype discussion on s, with some new features: theexistence part is obtained in presence of nonlineari-ties not necessarily bounded, and in the multiplicityresult it is not assumed a speed growth condition oran asymptotic condition, as it is usual in the litera-ture for these type of higher order problems.

The arguments used apply lower and upper solutionstechnique and topological degree theory.

An application to a continuous model of the humanspine, used in aircraft ejections, vehicle crash situa-tions and some forms of scoliosis, will be referred.

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Extremal points for an nth order three pointboundary value problem

Je↵rey NeugebauerEastern Kentucky University, USAje↵rey.neugebauer@eku.edu

We characterize first extremal points of an nth orderthree point boundary value by using a substitutionmethod and working with the 4th order problem.These results are then used to find positive solutionsof the nth order nonlinear problem.

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98 9th AIMS CONFERENCE – ABSTRACTS

Oscillation results for fourth order nonlinearmixed neutral di↵erential equations

Saroj PanigrahiUniversity of Hyderabad, Indiapanigrahi2008@gmail.comA. K. Tripathy, R. Basu

Oscillatory and asymptotic behaviour of a class ofnonlinear fourth order neutral di↵erential equationswith positive and negative coe�cients of the form

(r(t)(y(t) + p(t)y(t� ⌧))00)00+q(t)G(y(t� ↵))� h(t)H(y(t� �)) = 0,

and

(E) (r(t)(y(t) + p(t)y(t� ⌧))00)00+q(t)G(y(t� ↵))� h(t)H(y(t� �)) = f(t)

are investigated under the assumption

Z 1

0

tr(t)

dt =1

for various ranges of p(t). Using Scauder’s fixed pointtheorem, su�cient conditions are obtained for the ex-istence of positive bounded solutions of (E).

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Boundary value problems governing fluid flowand heat transfer over an unsteady stretchingsheet

Joseph PaulletPenn State Erie, USAjep7@psu.edu

This talk will consider two situations involving un-steady laminar boundary layer flow due to a stretch-ing surface in a quiescent viscous incompressiblefluid. In one configuration, the surface is imperme-able with prescribed heat flux, and in the other, thesurface is permeable with prescribed temperature.The boundary value problems governing a similarityreduction for each of these situations are investigatedand existence of a solution is proved for all relevantvalues of the physical parameters. Uniqueness ofthe solution is also proved for some (but not all)values of the parameters. Finally, a priori boundsare obtained for the skin friction coe�cient and localNusselt number.

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Existence analysis for nonlocal Sturm-Liouville boundary value problems

Jesus RodriguezNorth Carolina State Univerity, USArodrigu@ncsu.eduZach Abernathy

We establish conditions for the existence of solutionsto nonlinear di↵erential equations subject to nonlo-cal boundary conditions. The problems consideredare of the form

(p(t)x0(t))0 + q(t)x(t) + (x(t)) = G(x(t))

subject to the global boundary condition

↵x(0) + �x0(0) + ⌘1(x) = �1(x),�x(1) + �x0(1) + ⌘2(x) = �2(x). (1)

We assume : R! R is continuously di↵erentiable,p(t) > 0 and q(t) is real valued on [0, 1], p, p0, q arecontinuous on (0, 1), and the boundary conditions (1)are such that ↵2 + �2 6= 0, �2 + �2 6= 0. Further,G, ⌘1, ⌘2,�1 , and �2 shall be nonlinear operators de-fined on a function space.

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Applications of variational methods to anti-periodic boundary value problem for second-order di↵erential equations

Yu TianBeijing University of Posts and Telecommunications,China; Baylor University, USA(visitor scholar), Peo-ples Rep of Chinatianyu2992@163.comJohnny Henderson

We discuss the existence of multiple solutions toa second-order anti-periodic boundary value prob-lem by using variational methods and critical pointtheory. Furthermore, we get the existence of periodicsolutions for corresponding second-order di↵erentialequations. In constructing variational structure, weprove a fundamental lemma, which plays an impor-tant role in prove the critical point of functional isjust the solution of original problem.

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SPECIAL SESSION 22 99

Fractional boundary value problems with in-tegral boundary conditions

Min WangUniversity of Tennessee at Chattanooga, USAmin-wang@utc.eduJohn R. Graef, Lingju Kong, Qingkai Kong

The authors study a type of nonlinear fractionalboundary value problem with integral boundary con-ditions. By constructing an associated Green’s func-tion, applying spectral theory, and using fixed pointtheory on cones, they obtain criteria for the existence,multiplicity, and nonexistence of positive solutions.

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Existence, location and approximation resultsfor some nonlinear boundary value problems

Guangwa WangJiangsu Normal University, Peoples Rep of Chinawanggw7653@163.comMingru Zhou, Li Sun

The significance and importance of boundary valueproblems (BVPs) is well-known. The aim of thistalk is to present some results of existence, locationand approximation for some nonlinear BVPs. In ourstudy, the theory of di↵erential inequalities plays avery important role. More precisely, in the first part,we will present the existence and location criteriaof solutions for the general nonlinear system withthe general nonlinear boundary conditions. Here, weintroduce a new concept of bounding function pairand a method, which may be called simultaneousmodification. In the second part, using the gener-alized quasilinearization method, we will study twoclasses of second-order nonlinear BVPs with nonlocalboundary conditions. We will establish some su�-cient conditions under which corresponding mono-tone sequences converge uniformly and quadraticallyto the unique solution of the problem. The interest-ing point is that our boundary condition is nonlinearand nonlocal.

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Pengfei YuanSichuan University, Peoples Rep of Chinaypfxyz123@163.comShiqing Zhang

In this paper, we use a variant of the Benci-Rabinowitz Theorem to study the solutions of N-body-type problems with prescribed energies.

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